Apparatus and methods to find a position in an underground formation

ABSTRACT

Various embodiments include apparatus and methods related to finding a position in an underground formation. Apparatus and methods can include receiving signals from a receiver in an underground formation in response to signals generated from transmitting sources, each of the transmitting sources located at a known position; and processing the received signals, based on the signals generated from the transmitting sources, to determine the position of the receiver. A number of techniques can be applied to processing the received signal. Additional apparatus, systems, and methods are disclosed.

PRIORITY APPLICATIONS

This application is a continuation application of U.S. patentapplication Ser. No. 14/649,720, filed 4 Jun. 2015; which application isa U.S. National Stage Filing under 35 U.S.C. 371 from InternationalApplication No. PCT/US2012/072326, filed on 31 Dec. 2012, and publishedas WO 2014/105087 A1 on 3 Jul. 2014, which applications and publicationare incorporated herein by reference in their entirety.

TECHNICAL FIELD

The present invention relates generally to apparatus and method formaking measurements related to oil and gas exploration.

BACKGROUND

In drilling wells for oil and gas exploration, understanding thestructure and properties of the associated geological formation providesinformation to aid such exploration. Data to provide the information maybe obtained using sensors located in an underground formation at largedistances from the surface. Knowing the position of these sensors in theunderground formation can be used to formulate the information forexploration. Systems and techniques to determine the position of sensorsin the underground formation can enhance the analysis process associatedwith a drilling operation.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows an example placement of transmitters and a receiver, whichplacement can be used to determine the position of the receiver, inaccordance with various embodiments.

FIG. 2 shows a simulation setup for the analysis of the effect offrequency, in accordance with various embodiments.

FIGS. 3A-B show depth vs. voltage levels of received signals fordifferent frequencies for the simulation setup of FIG. 2, in accordancewith various embodiments.

FIGS. 4A-B show depth vs. voltage levels of received signals for varyingformation resistivities at a fixed operation frequency for thesimulation setup of FIG. 2, in accordance with various embodiments.

FIGS. 5A-B show depth vs. voltage levels of received signals for varyingformation resistivities at another fixed operation frequency for thesimulation setup of FIG. 2, in accordance with various embodiments.

FIG. 6 shows features of an example inversion scheme to determine theposition of a receiver in an underground formation, in accordance withvarious embodiments.

FIG. 7 shows features of an example of a constrained inversion scheme todetermine the position of a receiver in an underground formation, inaccordance with various embodiments.

FIG. 8 shows features of a simulation to verify an inversion scheme andto analyze the accuracy obtained in determining the position of one ormore sensors for different system configurations, in accordance withvarious embodiments.

FIG. 9 shows a simulation geometry for a positioning system with twox-directed transmitters at the surface, in accordance with variousembodiments.

FIGS. 10A-E show results of a Monte Carlo simulation for the simulationgeometry of FIG. 9, in accordance with various embodiments.

FIGS. 11A-E show results of a Monte Carlo simulation for the positioningsystem of FIG. 9 where a second receiver is used, whose position isconstrained with respect to a first receiver, in accordance with variousembodiments.

FIG. 12 shows a simulation geometry for a two transmitter positioningsystem, where one of the transmitters is underground, in accordance withvarious embodiments.

FIGS. 13A-E show results of a Monte Carlo simulation for the positioningsystem of FIG. 12, in accordance with various embodiments.

FIG. 14 shows a simulation geometry for a positioning system with fourtriad type transmitters, in accordance with various embodiments.

FIGS. 15A-E shows results of a Monte Carlo simulation for thepositioning system of FIG. 14, in accordance with various embodiments.

FIG. 16 shows a two-dimensional example with transmitters on the surfaceand a receiver underground to illustrate a method to find the positionof the receiver relative to the sources from the known orientations ofthe sources, in accordance with various embodiments.

FIG. 17 shows a two-dimensional example with transmitters on the surfaceand a receiver underground in which the receiver has a referencedirection, in accordance with various embodiments.

FIG. 18 shows a three-dimensional example with transmitters on thesurface and a receiver underground in which the receiver has noreference direction, in accordance with various embodiments.

FIGS. 19A-B show Monte Carlo simulation results using a semi-analyticalsolution for the positioning system shown in FIG. 9, in accordance withvarious embodiments.

FIG. 20 shows an electric field at the receiver, due to a magneticdipole, that is normal to the plane where receiver and transmitter arelocated, in accordance with various embodiments.

FIG. 21 shows a simulated system for electric field based positioningsystem, in accordance with various embodiments.

FIGS. 22A-C show Monte Carlo simulation results using an electric fieldbased positioning system of FIG. 21, in accordance with variousembodiments.

FIG. 23 depicts a block diagram of features of an example system to finda position in an underground formation, in accordance with variousembodiments.

DETAILED DESCRIPTION

The following detailed description refers to the accompanying drawingsthat show, by way of illustration and not limitation, variousembodiments in which the invention may be practiced. These embodimentsare described in sufficient detail to enable those skilled in the art topractice these and other embodiments. Other embodiments may be utilized,and structural, logical, and electrical changes may be made to theseembodiments. The various embodiments are not necessarily mutuallyexclusive, as some embodiments can be combined with one or more otherembodiments to form new embodiments. The following detailed descriptionis, therefore, not to be taken in a limiting sense.

In various embodiments, systems and methods to find the position of anunderground receiver can include locating the position of the receiver,or receivers, from measurements taken by the receivers and the knownpositions of the sources that generate the signals for the measurements.Sources can be used that are placed at known positions either at thesurface or below the surface of the earth with the receiver or receiverslocated underground. The position of underground receiver(s) from themeasurements of the signals, generated by a number of transmittingsources whose positions are known precisely, may be determined.

Electromagnetic type transmitting sources and receivers can be used insystems to determine a position underground. Such transmitting sourcescan include, but are not limited to, dipole transmitters, sourcesgenerating large distribution of current aboveground or near ground thatgenerate electromagnetic fields below ground, where the electromagneticfields measurable at the receiver, or other sources that can generate asignal measurable at a receiver deep in an underground formation.Dipoles of sources can be oriented in a direction perpendicular to thearea of interest, where the area of interest includes a receiving sourceto be located. This orientation can account for a null point along thedirection of the dipole. Transmitting sources can be realized by one ormore triad transmitters. A triad transmitter is a structure having threetransmitting sources at the same location, where the position ororientation of the three transmitting sources is different from eachother. The three transmitting sources of the triad can be mounted on thesame structure at a given location. Transmitting sources aboveground ornear ground can be operated to generate signals having a low frequencyto penetrate deeply underground such that the signals are measurable inan underground volume extending from a hundred feet to thousands feet indepth and from a hundred feet to thousands of feet across the depth.Alternatively, other types of transmitters used in oilfield explorationindustry, such as, but not limited to, acoustic sensors and seismicsensors, can be used in systems to determine a position underground. Thenumber of transmitting sources may include three or more transmittingsources. In an embodiment, three transmitting sources can be realized bya single triad transmitter.

The receiver or receivers can be controlled by electronics disposedunderground. In addition, a processing unit can be located downhole toanalysis the signals received by the receiver. The processing unit canbe realized by electronics integrated with the receiver, where theinformation concerning the known locations of the transmitting sensorsis stored with the electronics along with instructions to process thesignals. The processing unit can be realized by electronics disposed onthe structure on which the receiver is disposed and separated from thereceiver. The processing unit and receiver control located downhole canallow for automated geosteering. Alternatively, the processing unit canbe located at the surface, responsive to receiving the signals or dataregarding the signals from the receiver.

FIG. 1 shows an example embodiment of placement of transmitters and areceiver, which placement can be used to determine the position of thereceiver. In FIG. 1, three transmitters, denoted as Tx₁, Tx₂, and Tx₃,are illustrated, as an example, in different locations with respect to areceiving sensor, Rx₁. Tx₁ and Tx₂ are on surface 104, while Tx₃ isunderground inside a well different from the one in which Rx₁ isdisposed. This figure is a 2-dimensional (2D) figure, shown forillustration purposes, in which the transmitters line in the same plane.In various embodiments, transmitting sources used to locate the positionof a receiving sensor lie in a plane common to no more than twotransmitters and the receiver sensor. With the transmitting sensorssatisfying this condition, better resolution can be obtained in thereceived signals in a measurement process. In addition, the number oftransmitters, the number or receivers, or the number of transmitters andreceivers can be increased to improve resolution.

The number of transmitters and transmitter locations can also beoptimized using known optimization techniques, depending on theapplication. However, in the discussion of embodiments, the effects ofthe number of transmitters and their locations are analyzed usingnumerical modeling results. Receiver sensors and transmitters wereselected to be triads antennas, as illustrated in FIG. 1, in simulationsto improve inversion accuracy, although that is not necessary for theoperation of embodiments of methods to determine a position in anunderground formation. As shown in FIG. 1, transmitting sources Tx₁,Tx₂, and Tx₃ structured as three triads can provide 9 transmittingsources at three locations, with 9 positions or orientations. Receivingsensor, Rx₁, can also be structured as a triad receiving sensor havingthree receivers at one location with three positions or orientations.

Increasing the number of transmitting sources with each transmittingsource at a known location can increase the amount of information usedto determine the position of a receiver or receivers in undergroundformations. In addition, the transmitting sources are not limited tousing the same type of transmitting source. For example, arrangementscan include two triad transmitters among three or more transmittersdistributed over a significantly large region. Other arrangements caninclude a transmitting source structured as a circuit distributed over asignificantly large region on the surface or near the surface. Thecircuit can include a closed loop having a current-carrying wire, wherethe current-carrying wire is at a known position and thecurrent-carrying wire is arranged along a straight-line path such thatsignals received at the receiver from the closed loop are negligiblefrom portions of the closed loop that follow a path different from thestraight-line path. The signal at the receiver can be primarily providedby this single current-carrying wire with the other portions of thecircuit that close the loop located at such distances from the receiverthat signals from these other portions are effectively attenuated priorto the receiver. Alternatively, a transmitting source can be structuredas a circuit having a closed loop with a number of current-carryingwires with each current-carrying wire being at a known position andarranged along a straight-line path such that signals received at thereceiver from the closed loop are negligible from portions of the closedloop that follow a path different from these respective straight-linepaths. The received signals at the receiver can be processed based on amodel of the number of current-carrying wires and their correspondingstraight-line paths.

Low frequency electromagnetic waves can penetrate deeply below thesurface of the earth. By using low frequency sources (f<10 Hz), thefields generated by the sources will be measurable at positions burieddeeply underground. In an embodiment, low frequency sources having afrequency less than or equal to 50 Hz can be used. A receiver placed ina borehole under the surface can measure the signals generated by thesources. These signals from each one of the sources can be processed tofind the distance, orientation, or both distance and orientation. Invarious embodiments, position determination at depths as large as 10,000meters may be performed.

A first consideration includes the effect of frequency on the signalsthat penetrate the formation. As frequency increases, attenuationunderground increases such that, at higher frequencies, attenuation ismore severe and could reduce signals below the noise level. Anotherfactor to consider is that higher frequencies are also more sensitive tothe formation, which can significantly affect the received signal. Thatis, at higher frequencies, the solution for the position of areceiver(s) would be sensitive to the parameters of the complex and inmost cases not accurately known formation information. Exampleembodiments of methods to determine a position underground can beperformed under a single frequency operation. However, in otherembodiments, features of methods to determine a position underground canbe performed under a multi-frequency operation.

FIG. 2 shows a simulation setup for the analysis of the effect offrequency. To analyze the effect of frequency, the variation of thesignal level with depth is computed as a function of frequency when asingle transmitter is present. Transmitter and receiving sensor weresimulated as coil antennas with their normal parallel to the radialdirection of earth. Henceforth, the axis normal to the surface of earthwill be denoted as z-axis. With this convention, the simulated caseshows the ZZ-coupling. Other orthogonal components (XX-coupling andYY-coupling) show similar characteristics and are not shown.

In this simulation, receiving sensor Rx is directly below thetransmitter Tx. For illustration purposes, both the receiver andtransmitter coils are assumed to have a unit area with 400 turns eachand the transmitter is assumed to carry 25 A current. Design parameterscan vary as dictated by engineering concerns for a given application.However, to be able to transmit signal to such great depths, transmittedpower level should be high and the receiving sensor structured as ahighly sensitive receiver. This concern for power level may also makeputting transmitters on the surface 204 more practical than cases wheresome or all of the transmitters are under ground. Formation parametersused in the simulation are also shown in FIG. 2. A somewhat worst-casescenario with a conductive formation of resistivity, R_(f), equal to 1Ω-m was considered. Relative permittivity (ε_(r)) and permeability(μ_(r)) were selected as 5 and 1, respectively.

FIGS. 3A-B show depth vs. voltage levels of received signals fordifferent frequencies for the simulation setup of FIG. 2. FIG. 3A showsthe change of the absolute value of the voltage with depth for fourdifferent frequencies, and FIG. 3B shows the change of the phase of thevoltage with depth for the four different frequencies. Curves 342, 344,346, and 348 show depth as a function of the absolute value of voltageat frequencies of 0.01 Hz, 0.1 Hz, 1 Hz, and 10 Hz, respectively. Curves352, 354, 356, and 358 show depth as a function of the phase of thevoltage at frequencies of 0.01 Hz, 0.1 Hz, 1 Hz, and 10 Hz,respectively. For 1 Hz and 10 Hz, signal quickly attenuates. Wrap aroundin the phase can also be seen which can make the inversion difficult. Incomparison, as the frequency gets lower, signal attenuation becomes lessof a problem. However, for these lower frequencies, initial strength ofthe signal is already low. Thus, even for 0.01 Hz and 0.1 Hz, thevoltage level goes to as low as 10 femtoVolts at 10,000 m for thesimulated transmitter and receiver configurations. Results suggest thatfor the parameters used there is little improvement in attenuation forfrequencies lower than 0.1 Hz. Thus, in other simulations discussedherein, the frequency of operation was assumed to be 0.1 Hz.

FIGS. 4A-B show depth vs. voltage levels of received signals for varyingformation resistivities at a fixed operation frequency. To analyze theeffect of formation resistivity, the same setup shown in FIG. 2 was usedwith the frequency set to a constant 0.1 Hz and the formationresistivity varied between 0.1 Ω-m to 100 Ω-m. Curves 442, 444, 446, and448 show depth as a function of the absolute value of voltage atformation resistivities of 0.1 Ω-m, 1 Ω-m, 10 Ω-m, and 100 Ω-m,respectively. Curves 452, 454, 456, and 458 show depth as a function ofthe phase of the voltage at formation resistivities of 0.1 Ω-m, 1 Ω-m,10 Ω-m, and 100 Ω-m, respectively. Effect of formation resistivity onthe received signal can be seen to be small except for extremelyconductive formations. Thus, this effect can be neglected, or it can beeliminated using a basic correction scheme. Nevertheless, for examplesdiscussed herein, formation resistivity was assumed to be exactly known.

FIGS. 5A-B show depth vs. voltage levels of received signals for varyingformation resistivities at another fixed operation frequency, inaccordance with various embodiments. The same setup shown in FIG. 2 wasused with the frequency set to a constant 0.01 Hz and the formationresistivity varied between 0.1 Ω-m to 100 Ω-m. Curves 542, 544, 546, and548 show depth as a function of the absolute value of voltage atformation resistivities of 0.1 Ω-m, 1 Ω-m, 10 Ω-m, and 100 Ω-m,respectively. Curves 546 and 548 overlap such that the differences arenot discernible at the scales of FIG. 5A. Curves 552, 554, 556, and 558show depth as a function of the phase of the voltage at formationresistivities of 0.1 Ω-m, 1 Ω-m, 10 Ω-m, and 100 Ω-m, respectively.Results for 0.01 Hz shown in FIG. 5 exhibit very little dependence onthe formation resistivity for the depth range considered. However, sucha low frequency may cause difficulties in an implementation of thesystem hardware.

FIG. 6 shows features of an example inversion scheme to determine theposition of a receiver in an underground formation. This inversionscheme demonstrates how the position of the receiving sensor may bedetermined using an array of transmitters at previously known locations.At 610, the measurement of signals due to N different transmitters atthe receiver is acquired. These signals are combined into a columnvector, denoted as V, at 620. Although a single receiver relative to Ntransmitters is discussed at 610 and 620, more complicated measurementscan be considered in a similar or identical manner. For example, ifreceivers or transmitters are multi-component, each individual entry(V_(TxRx)) becomes a vector with individual components as the element ofthe vector. Examples of such receivers and transmitters include triadreceivers and triad transmitters. If there are multiple frequencies,results of these measurements may be appended to the measurement vectorand so on. Once this voltage is obtained, it may be further processeddepending on the application. For example, if signal from one of thetransmitters is too strong compared to the others, amplitudes ofreceived signal from different transmitters may be normalized toascertain that weight of each transmitter in the inversion is same.

In the inversion scheme, the determination of the position and thedirection of the receiving sensor(s) is the object of interest. Thus,parameters of interest are denoted as the location of the receiversensor (x, y, z), its azimuth (θ), and its elevation angle (ϕ). At 630,an initial guess of the location and direction parameters (x′, y′, z′,θ′, ϕ′) is made. The signal corresponding to an initial guess of thelocation and direction parameters (x′, y′, z′, θ′, ϕ′) is simulatedusing a forward model, which is denoted as V′ at 640. As in everyinversion scheme, an accurate forward model that relates parameters tobe inverted to the measured signal is used in this method.

At 650, the norm of the difference between V and V′ is compared to athreshold. If the norm of the difference between V and V′ is lower thana predetermined threshold, the processing may stop and the processedparameters (x′, y′, z′, θ′, ϕ′) may be deemed to be accurateapproximations to the true parameters (x, y, z, θ, ϕ), at 660. Otherconvergence criteria may also be applied in this step.

If convergence is not satisfied, an iteration number can be increased byone, at 670. To prevent, for example, infinite simulations for caseswhere no solution below the threshold is possible such as at highlynoisy environments, or to restrict the simulation time, the number ofiterations may be compared with a previously set maximum iterationnumber, at 680. If the maximum number of iterations is reached, theprocessing may stop with the latest guess, or a previous guess thatminimized the error, returned as the answer, at 685. Otherwise, theparameter guess vector is updated at 690, V′ is simulated again, at 640,based on this guess and the above process of comparing the processsignal with the measured signal and subsequent comparisons can berepeated. The update of the guess vector may be based on the calculationof a gradient that minimizes the error.

Alternative inversion schemes may be used with equal success. Suchinversion schemes can include using a lookup table. Another alternativeinversion scheme can include applying a brute force search method thattries a large number of possible input combinations and selects the onethat minimizes the error between the measured data and the forwardmodel. Alternative inversion schemes are not limited to thesealternatives, but may include other alternative inversion schemes orcombinations thereof.

FIG. 7 shows features of an example of a constrained inversion scheme todetermine the position of a receiver in an underground formation, inaccordance with various embodiments. A constraint may be applied as oneof a number of different techniques may be employed to reduce the errorin inversion. One such technique is the addition of a second sensorwhose position relative to the first sensor is exactly known. Althoughthese two sensors will have to be close to each other in the electricalsense, thus providing little independent information, the fact that thenoise at separate sensors should be mostly independent will improve theinversion accuracy. Inversion in this case can be similar to theinversion with a single receiver associated with FIG. 6.

At 710, the measurement of signals due to N different transmitters atthe two receivers is acquired. These signals are combined into a columnvector, denoted as V, at 720, providing twice as many components as themeasured signal in methods related to FIG. 6. These measured signals canbe acquired and processed in a manner similar to the variations ofprocessing measured signals with respect to FIG. 6.

In the inversion scheme, the determination of the position and thedirection of the receiving sensor(s) is the object of interest. Thus,parameters of interest are denoted as the location of the receiversensor (x, y, z), its azimuth (θ), and its elevation angle (ϕ). At 730,an initial guess for the position and orientation parameters (x′, y′,z′, θ′, ϕ′) is made of one of the sensors. Since the exact location ofthe second sensor is known with respect to the first sensor, the guessfor its position and orientation may be calculated based on the firstguess, at 735. The signal corresponding to the initial guess of thelocation and direction parameters of the two receivers is simulatedusing a forward model, which is denoted as V′ at 740. An accurateforward model, which relates parameters to be inverted to the measuredsignal, can be used in this method.

At 750, the norm of the difference between V and V′ for the tworeceivers is compared to a threshold. If the norm of the differencebetween V and V′ is lower than a predetermined threshold, the processingmay stop and the processed parameters (x′, y′, z′, θ′, ϕ′) may be deemedto be accurate approximations to the true parameters (x, y, z, θ, ϕ), at760. Other convergence criteria may also be applied in this step.

If convergence is not satisfied, an iteration number can be increased byone, at 770. To prevent, for example, infinite simulations for caseswhere no solution below the threshold is possible such as at highlynoisy environments, or to restrict the simulation time, the number ofiterations may be compared with a previously set maximum iterationnumber, at 780. If the maximum number of iterations is reached, theprocessing may stop with the latest guess, or a previous guess thatminimized the error, returned as the answer, at 785. Otherwise, theparameter guess vector is updated at 790 with the parameters for theother receiver updated, since the exact location of the second sensor isknown with respect to the first sensor. V′ is simulated again, at 740,based on these updated guesses and the above process of comparing theprocess signal with the measured signal and subsequent comparisons canbe repeated. The update of the guess vector may be based on thecalculation of a gradient that minimizes the error.

FIG. 8 shows features of a simulation to verify an inversion scheme andto analyze the accuracy obtained in determining the position of one ormore sensors for different system configurations. These simulations wereconducted as Monte-Carlo simulations. At 810, the process begins withthe true position/orientation vector (x, y, z, θ, ϕ). In the Monte Carlosimulations, an ideal signal is found, at 820, using the forward modelcorresponding to the position/orientation vector (x, y, z, θ, ϕ). Tosimulate the environmental and system noises and other measurementuncertainties, a random noise, η_(i), can added to the ideal signal,V_(ideal), to create the “measured” signal 610 of FIG. 6 and 710 of FIG.7. The noises added to each row of V_(i), that is each channel, areselected to be independent of each other. Here, the subscript irepresents the iteration number of the Monte Carlo simulation. A uniformdistribution between (−0.5 and 0.5) is used to create the random noise.The amplitude of this random noise is then scaled, and added in amultiplicative manner to the original signal as follows:V _(i,j) =V _(ideal,j)×(1+u(−0.5,0.5)/SNR)  (1)In equation (1), j represents the index of a row of vectors V_(i) andV_(ideal), u(−0.5,0.5) represents a uniform random noise taking itsvalues between −0.5 and 0.5, and SNR is the scaling factor thatrepresents a signal-to-noise ratio. In the simulations, SNR was selectedto be 50. V_(i) is then inverted to produce the guess (x_(i), y_(i),z_(i), θ_(i), ϕ_(i)) for iteration I, stored at 840. The above processis repeated N times, using a counter at 850, to be able to accuratelyanalyze the inversion performance for varying noise. Number ofiterations (N) was selected as 100 in the simulations.

FIG. 9 shows a simulation geometry for a positioning system with twox-directed transmitters at the surface. A reference coordinate system interms of (x, z) with z being in the direction from surface 904 isindicated in FIG. 9 with origin (0, 0) at x-axis and z-axis shown. Thepositioning system of this example consists of two identicaltransmitters, Tx₁ and Tx₂, which are x-directed magnetic dipoles. Thesetransmitters are located at positions of (x, y, z)=(1000, 0, 0) metersand (3000, 0, 0) meters with respect to the origin where x, y and z arepositions in the x-direction, y-direction, and z-direction with respectto the reference coordinate system. The receiver, Rx, is a triad ofmagnetic dipoles. Its position is selected to be (2300, 0, z_(rec)),where z_(rec) is the true vertical depth (TVD) and changed from 100 m to10,000 m in 100 m steps to emulate the descent of receiver into theground. Rx is assumed to have an elevation angle of 70° and an azimuthangle of 50°. Formation is assumed to have a resistivity of R_(f)=1 Ω-m,relative dielectric permittivity of ε_(r)=5 and a relative magneticpermeability of μ_(r)=1. Although Rx, Tx₁, and Tx₂ lie on the same planefor this particular example, inversion does not incorporate thisinformation. In other words, it is assumed that Rx may lie anywhere inthe 3-dimensional space. In addition, drilling rig 902, Tx₁, and Tx₂ lieon the surface 904 for illustrative purposes, and formation 901 isassumed to be homogeneous.

Voltage received at the Rx sensor for this system is a vector with sixcomponents. With a goal to solve the position and orientation of Rx,there are five unknowns in the problem. Thus, the solution isoverdetermined. With similar reasoning, it can be seen that even with asingle transmitter and a single triad receiver, positioning is possibleif orientation of the sensor is known via other means. For example, theorientation of Rx may be determined with the use of inclinometers.

FIGS. 10A-E show results of a Monte Carlo simulation for the simulationgeometry of FIG. 9. These results are with 100 repetitions at each depthpoint, z_(rec), of Rx. Curves 1042, 1052, 1062, 1072, and 1082 indicatethe mean error between the sensor position and the simulations. Curves1044, 1054, 1064, 1074, and 1084 show plus one standard deviation of theerror from the mean. Curves 1046, 1056, 1066, 1076, and 1086 show minusone standard deviation of the error from the mean. In the Monte Carlosimulations, if the mismatch between the measured voltage and thevoltage obtained using the inverted parameters is above a threshold,that particular inversion is discarded. This is akin to the real-timesituation where an inversion would be deemed useless if the voltagecalculated from the inverted parameters has a large difference withrespect to the measured voltage. It can be seen from these results thatthe orientation of the sensor may be accurately determined even at largedepths. Position determination is less accurate but the mean errorgenerally stays within 5 meters for each position component.

FIGS. 11A-E show results of a Monte Carlo simulation for the positioningsystem of FIG. 9 where a second receiver is used, whose position isconstrained with respect to the first receiver. In the simulation, thesecond receiver has a location constrained with respect to the firstreceiver such that the second receiver is 10 m below the first one inthe tool axis and the orientation of the two receivers are same. Curves1142, 1152, 1162, 1172, and 1182 indicate the mean error between thesensor position and the simulations. Curves 1144, 1154, 1164, 1174, and1184 show plus one standard deviation of the error from the mean. Curves1146, 1156, 1166, 1176, and 1186 show minus one standard deviation ofthe error from the mean. A slight improvement in inversion performancecan be observed with the additional knowledge obtained from this secondreceiver.

FIG. 12 shows a simulation geometry for a two transmitter positioningsystem where one of the transmitters is underground. This system issubstantially the same as the geometry shown in FIG. 9, except one ofthe transmitters, Tx₁, is located underground at point (1000, 0, 1000)with respect to the origin with transmitter, Tx₂, on the surface at(3000, 0, 0). Receiver, Rx, at (2300, 0, z_(rec)) is assumed to have anelevation angle of 70° and an azimuth angle of 50°. Formation is assumedto have a resistivity of R_(f)=1 Ω-m, relative dielectric permittivityof ε_(r)=5 and a relative magnetic permeability of μ_(r)=1.

FIGS. 13A-E show results of a Monte Carlo simulation for the positioningsystem of FIG. 12. Curves 1342, 1352, 1362, 1372, and 1382 indicate themean error between the sensor position and the simulations. Curves 1344,1354, 1364, 1374, and 1384 show plus one standard deviation of the errorfrom the mean. Curves 1346, 1356, 1366, 1376, and 1386 show minus onestandard deviation of the error from the mean. Results are similar tothe case with both transmitters on the surface. In fact, a slightimprovement in inversion performance can be observed, which can beattributed to the fact that the location of the transmitters in thepositioning system of FIG. 12 better span the space. Thus, informationobtained from these two transmitters is more independent.

FIG. 14 shows a simulation geometry for a positioning system with fourtriad type transmitters. The example positioning system, which wassimulated, consists of triad type transmitters, Tx₁, Tx₂, Tx₃, and Tx₄,at locations (1000, 0, 1000), (2000, 1000, 0), (2000, −1000, 0), and(3000, 0, 0), respectively. A deployed system can include transmittersat different locations and can include an increased number oftransmitters. For the simulation, receiver, Rx, is a triad type receiverat a position of (2300, 0, z_(rec)), where zrec represents the truevertical depth, and Rx has 70° elevation angle and 50° azimuth angle.Formation is assumed to have a resistivity of R_(f)=1 Ω-m, relativedielectric permittivity of ε_(r)=5 and a relative magnetic permeabilityof μ_(r)=1. Simulation results indicate that further improvements may beobtained by using triad type transmitters and increasing the number oftransmitters.

FIG. 15A-E shows results of a Monte Carlo simulation for the positioningsystem of FIG. 14. Curves 1542, 1552, 1562, 1572, and 1582 indicate themean error between the sensor position and the simulations. Curves 1544,1554, 1564, 1574, and 1584 show plus one standard deviation of the errorfrom the mean. Curves 1546, 1556, 1566, 1576, and 1586 show minus onestandard deviation of the error from the mean. Standard deviation oferror is cut almost in half compared to the system depicted in FIG. 12.

Methods other than using full inversion can be implemented to find theposition of a receiver sensor or sensors in an underground formation.These other methods provide semi-analytical formulations to find theposition. For example, once the angular position of the sources is foundfrom the measurements, then by geometrical identities the position ofthe receiver can be found. Using the angular information alone may beadvantageous in some situations if the magnitude of the signal from thesources could be affected by parameters other than the distance. Forexample, dispersion or refraction effects of the medium between thesource and the receiver can affect the magnitude of the signal from thesources. Once the direction of the sources has been found, such as byusing inversion methods, the angles (θ_(source), ϕ_(source)) for eachsource are known. Once the orientations of the different sources at thesurface or inside the formation are found, the position of theunderground receiver relative to the sources can be deduced by geometricidentities. Additional information can be obtained from the direction ofthe fields of each transmitter antenna. The information about thedirection of the fields can help reduce the error in the determinationof receiver position.

FIG. 16 shows 2-dimensional example with transmitters, Tx₁, Tx₂, Tx₃, onthe surface and the receiver, Rx, underground to illustrate a method tofind the position of the receiver relative to the sources from the knownorientations of the sources. The position of Rx underground can be foundusing semi-analytical formulations by measuring the angular orientationof the sources at the surface and solving the trigonometric problemassociated with the transmitters and receiver, where the position of thesources at the surface is known precisely. Additional information can beobtained from the direction of the fields of each transmitter antenna.The information about the direction of the fields can help reduce theerror in the determination of receiver position. However, in thisexample, knowledge only of the direction of the antennas is assumed andnot of the electric fields.

In this 2D example, shown in FIG. 16, all receiver and transmitters arelocated on the same plane. In this case, Rx does not have any referenceto measure the angle of the sources. The measurements are the angles αand β between the directions of the sources. From the trigonometricidentities of the cosine theorem, the following three equations can bederived:d ₁ ² =a ² +b ²−2ab cos(α)  (2)d ₂ ² =b ² +c ²−2bc cos(β)  (3)(d ₁ +d ₂)² =a ² +c ²−2ac cos(α+β)  (4)In equations (2), (3) and (4) the unknowns a, b and c can be obtained asfunctions of d₁ and d₂, which are the distances between the sources onthe surface. Distance d₁ is the distance between Tx₁ and Tx₂, anddistance d₂ is the distance between Tx₂ and Tx₃. These distances can beknown with high precision. To solve for the position of Rx, it can beassumed that Tx₁ has position (0, 0). The cosine theorem can be appliedagain to obtain:b ² =a ² +c ²−2ab cos(δ),  (5)from which the angle δ can be obtained. The coordinates of the receiverposition can be evaluated as x=a cos(δ) and y=a sin(δ).

FIG. 17 shows a two-dimensional example with transmitters Tx₁ and Tx₂ onthe surface and a receiver Rx underground in which the receiver has areference direction. Tx₁ and Tx₂ are separated by a distance d₁. Thereference known to Rx can be the direction of gravity, which pointsapproximately towards the center of the earth. The direction of gravityis known and a plane 1706 perpendicular to the direction of gravity canbe constructed. The directions of the sources can be referenced to theplane perpendicular to gravity. The angles α and β represent thedirections of the sources. Assuming that the plane 1706 perpendicular tothe direction of gravity and the surface of earth are parallel to eachother, then, by a geometric theorem, Φ₁=β and Φ₂=α. With two internalangles Φ₁ and Φ₂ and the length d₁ known, all sides and angles of atriangle can be solved. Thus, in this example with a reference directionprovided, only two sources are needed for this semi-analyticalformulation.

The number of sources needed to find the position underground depends onhow many directional references are available. If gravity gives areference direction to the center of the earth and the local magneticfield orientation is known, providing a second reference direction, thenthe position of a receiver underground can be found with only twosources on the surface, assuming knowledge only of the angular positionof the sources without information about the distance. Other methodsinclude using a 3D situation with respect to transmitters and areceiver, without a given reference, in a manner similar to theabovementioned method in the 2D case, which makes use of the cosinetheorem.

FIG. 18 shows a three-dimensional example with transmitters on thesurface and a receiver underground in which the receiver has noreference direction, in accordance with various embodiments. S1, S2, andS3 are locations on the surface of three transmitters. The receiver isat location O. The distances d₁ between S1 and S2, d₂ between S2 and S3,and d₃ between S1 and S3 are known precisely. Angles α, β, and δ can bemeasured. From FIG. 18 the following equations hold:d ₁ ² =a ² +b ²−2ab cos(α)  (6)d ₂ ² =b ² +c ²−2bc cos(β)  (7)d ₃ ² =a ² +c ²−2ac cos(δ)  (8)where the unknowns a, b and c can be found. From the known positions ofthe sources and the solved a, b and c, the position of the receiver at Ocan be found in these semi-analytical formulations. If the source is atriad and the direction of the source dipole is known, then there ismore information available because the direction of oscillation of thesource field provides extra information. In addition, to be able todistinguish different sources, each source could use a differentfrequency. The use of more sources is convenient because it can improvethe accuracy of the positioning.

Simulations can be applied to semi-analytical approaches. Using thepositioning system shown in FIG. 9A, a simulation example can bepresented using 2D semi-analytical formulations. For this 2D example, itwas assumed that the receiver is known to lie on the same plane with thetwo transmitters. Then, angles α and β, depicted in FIG. 17, can befound as the arc tangent of the ratio of vertical and horizontaldistances from each transmitter to the receiver. These distances can befound by using the received data due to a single transmitter rather thanusing the previously discussed inversion schemes. Such an approach canbe conducted by comparing measured signals with calculated signalssimilar to the inversion schemes discussed before. The results of asemi-analytical approach may be more useful in cases where gainfluctuations in receiver and transmitter are an issue or indispersive/refractive media.

In some other applications, if transmitter and receiver are both triadswith known orientations, they can be rotated in a way to obtain twodipoles where the received signal is zero. This is only possible if thereceiver is at infinity, which can be discarded in practicalapplications; or if the transmitting and receiving dipoles lie parallelto the line connecting them, which provides the angular information.Other variations and combinations of the aforementioned methods may alsobe used to find the angular information.

FIGS. 19A-B shows Monte Carlo simulation results using a semi-analyticalsolution for the positioning system shown in FIG. 9. The receiverposition in the x-plane and the z-plane were found using angularinformation. True vertical depth is changed between 100 and 10,000meters in 100 meter steps, and 100 repetitions of simulations wereperformed at each step for an SNR value of 50. Curves 1942 and 1952indicate the mean error between the sensor position and the simulations.Curves 1944 and 1954 show plus one standard deviation of the error fromthe mean. Curves 1946 and 1956 show minus one standard deviation of theerror from the mean. Results have slightly less accuracy than the onesshown in FIG. 10, since angular information is obtained using theinteraction between just a single transmitter and a single receiver.

In various embodiments, methods to find a position in an undergroundformation can include electric-field based positioning. In the previousexamples, both transmitters and receivers were assumed to be magneticdipoles. Thus, receivers measured the magnetic fields. If electricfields at the receivers are measured instead, a different approach maybe used to obtain the position of the receiver.

FIG. 20 shows an electric field at the receiver, due to a magneticdipole, that is normal to the plane where receiver and transmitter arelocated. The electric field of a magnetic dipole only has acircumferential (ϕ) component. Thus, the electric field, E(x_(rec),y_(rec), z_(rec)) at the receiver is normal to the plane where thetransmitter T_(x)(x₀, y₀, z_(rec)) and receiver lie. This plane may thusbe defined by the following formula:n _(x0)(x−x ₀)+n _(y0)(y−y ₀)+n _(z0)(z−z ₀)=0  (9)In equation (9), n_(x0), n_(y0) and n_(z0) represent the x, y, and zcomponents of a unit vector that has the same direction with theelectric field and (x₀,y₀,z₀) is the transmitter position. If there arethree such linearly independent equations for three differenttransmitters (in other words if planes obtained from equation (9) arenot the same for two or more transmitters), the independent equationsmay be solved to obtain the receiver position as shown in equation (10).The vector (n_(xi), n_(yi), n_(zi)) is the unit vector parallel to theelectric field at the receiver produced by transmitter i at location(x_(i), y_(i), z_(i)), where i=0, 1, 2.

$\begin{matrix}{\begin{bmatrix}x \\y \\z\end{bmatrix} = {\begin{bmatrix}n_{x\; 0} & n_{y\; 0} & n_{z\; 0} \\n_{x\; 1} & n_{y\; 1} & n_{z\; 1} \\n_{x\; 2} & n_{y\; 2} & n_{z\; 2}\end{bmatrix}^{- 1}\begin{bmatrix}{{n_{x\; 0}x_{0}} + {n_{y\; 0}y_{0}} + {n_{z\; 0}z_{0}}} \\{{n_{x\; 1}x_{1}} + {n_{y\; 1}y_{1}} + {n_{z\; 1}z_{1}}} \\{{n_{x\; 2}x_{2}} + {n_{y\; 2}y_{2}} + {n_{z\; 2}z_{2}}}\end{bmatrix}}} & (10)\end{matrix}$For practical applications, it is straight forward to satisfy theindependence requirement in a volume of interest. For example, if allthree transmitters lie on a flat surface apart from each other, and thereceiver is not on this surface, planes obtained by equation (9) willalways intersect at the point where the receiver is located.

In practical applications, noise will affect the accuracy of theresults. In those cases, it may be desired to add additional informationto improve accuracy. Additional information can include using additionaltransmitting sources, each at a known position. For these cases, thematrix of unit vectors may not be a square matrix. Thus, apseudo-inverse of the matrix should be used instead in equation (10). Noiterative inversion is applied in this approach; thus, results areobtained much faster than the inversion approach. However, orientationof the receiver sensor must be known accurately via other means.

FIG. 21 shows a simulated system for electric field based positioningsystem. A reference coordinate system in terms of (x, y, z) with z beingin the direction from surface 2104 is indicated in FIG. 21 with origin(0, 0, 0) at x-axis, y-axis, and z-axis shown. The positioning system ofthis example has three transmitters: Tx₁, a x-directed magnetic dipoleat (1000, 0, 0); Tx₂, a y-directed magnetic dipole at (3000, 0, 0); andTx₃, a z-directed magnetic dipole at (2000, −1000, 0). Receiver Rx, atriad of electric dipoles, is at (2300, 600, z_(rec)), where z_(rec)represents the TVD. As before, Monte Carlo simulations are repeated 100times at each depth step as TVD is changed from 100 m to 10,000 m insteps of 100 m and SNR is taken as 50. Rx is assumed to have anelevation angle of 70° and an azimuth angle of 50°. Formation is assumedto have a resistivity of R_(f)=1 Ω-m, relative dielectric permittivityof ε_(r)=5 and a relative magnetic permeability of μ_(r)=1.

FIGS. 22A-C show Monte Carlo simulation results using the electric fieldbased positioning system of FIG. 21. The Monte Carlo simulation resultsare for errors in x-position, y-position, and z-position of the receivershown in FIG. 21. Curves 2242, 2252, and 2262 indicate the mean errorbetween the sensor position and the simulations. Curves 2244, 2254, and2264 show plus one standard deviation of the error from the mean. Curves2246, 2256, and 2266 show minus one standard deviation of the error fromthe mean. Accuracy is not as good as the inversion approach, but still areasonable approximation to the true receiver location is obtained.

Based on the duality theorem, electric dipoles can be used in theexamples discussed herein. For example, if transmitters are electricdipoles, a magnetic field will be normal to the plane containing thereceiver and transmitter locations instead of the electric field. Thus,the method described herein may be used by measuring the magnetic fieldsat the receiver.

In various embodiments, features of a method to locate a receiverdownhole comprise: receiving signals from a receiver in an undergroundformation in response to signals generated from three or moretransmitting sources, each of the three or more transmitting sourceslocated at a known position, at least one transmitting source of thethree or more transmitting sources separated from and mounted on astructure different from at least one other transmitting source of thethree or more transmitting sources; and processing the received signals,using an inversion process based on the signals generated from the threeor more transmitting sources, to determine the position of the receiver.The processing of the signals to determine the position of the receivercan be conducted downhole. Downhole processing can be conducted usingelectronics integrated with the receiver, where the informationconcerning the known locations of the transmitting sensors is storedwith the electronics along with instructions to process the signals. Thedownhole processing can be located using electronics disposed on thestructure on which the receiver is disposed and separated from thereceiver. The downhole processing can allow for automated geosteering.Alternatively, the processing unit can be conducted at the surface inresponse to receiving the signals or data regarding the signals from thereceiver.

Features of the method can include controlling the three or moretransmitting sources including a transmitting source that has acurrent-carrying wire of a closed loop of a circuit, thecurrent-carrying wire being at a known position and the current-carryingwire is arranged the current-carrying wire arranged along astraight-line path such that signals received at the receiver from theclosed loop are negligible from portions of the closed loop that followa path different from the straight-line path. Features of the method caninclude controlling the three or more transmitting sources including atransmitting source that has a number of current-carrying wires forminga closed loop of a circuit, each of the number of current-carrying wiresbeing at a known position and arranged along a straight-line path suchthat signals received at the receiver from the closed loop arenegligible from portions of the closed loop that follow a path differentfrom the straight-line paths; and processing the signals based on amodel of the number of current-carrying wires and their correspondingstraight-line paths. Controlling the three or more transmitting sourcescan include controlling at least three dipole transmitters. Controllingthe three or more transmitting sources can include controlling a sourcegenerating large distribution of current aboveground or near ground thatgenerate electromagnetic fields below ground, the electromagnetic fieldsmeasurable at the receiver, the large distribution of current being at aknown position. The three or more transmitting sources include one ormore transmitting sources located aboveground. The three or moretransmitting sources can include a transmitter in a well. The well canbe different from a well in which the receiver is located or the wellcan be the well in which the receiver is located. The three or moretransmitting sources can include no more than two transmitters in aplane that contains the receiver. Controlling the three or moretransmitting sources can include conducting various combinations ofthese embodiments of features to control the three or more transmittingsources.

Features of the method can include generating at least one signal of thegenerated signals from a transmitting source aboveground or near ground,the signal having a low frequency to penetrate deeply underground suchthat the signal is measurable in an underground volume extending from ahundred feet to thousands feet in depth and from a hundred feet tothousands of feet across the depth. The method can include operating thethree or more transmitting sources sequentially such that only one ofthe three or more transmitting sources is on at one time period. Themethod can include operating each of the transmitting sources at afrequency less than about 50 Hz.

Using an inversion process can include: generating values of a signalexpected at the receiver from each of the transmitting sources;generating a difference between the signal expected and the signalreceived from the receiver; when the difference is less than athreshold, selecting values of coordinates for the receiver, as theposition of the receiver, that generated the signal expected at thereceiver for which the difference is less than the threshold; and whenthe difference is greater than the threshold, generating new values of asignal expected at the receiver and determine if a difference betweenthe new values and the signal received from the receiver is less thanthe threshold. Generating values of the signal expected at the receivercan include using an estimate of the position of the receiver with aforward model. Generating values of the signal expected at the receivercan include using an estimate of the position of the receiver with alookup table.

Using an inversion process can include: generating an estimate of theposition of the receiver, the receiver taken as a first receiver;generating an estimate of each position of one or more other receivers,each of the one or more other receivers having a known position withrespect to the first receiver; generating values of signals expected atthe first receiver and at the one or more other receivers from each ofthe transmitting sources; generating a difference between the values ofthe signals expected and a combination of the signal received at thefirst receiver and signals received at the one or more other receivers;when the difference is less than a threshold, selecting values ofcoordinates of the first receiver, as the position of the firstreceiver, that generated the signals expected at the first receiver forwhich the difference is less than the threshold; and when the differenceis greater than the threshold, generating a new estimate of the positionof the first receiver, if the inversion process is within a maximumiteration.

Using an inversion process can include: generating sets of values of asignal expected at the receiver, each set generated from a differentestimate of the position of the receiver; generating differences betweenthe values of the signal expected and the signal received from thereceiver for each set; and selecting the estimate that minimizes errorin the difference between the values of the signal expected and thesignal received from the receiver. Generating the sets of values ofsignals expected at the receiver can include using a forward model witheach of the estimates.

In various embodiments, features of a second method to locate a receiverdownhole comprise: receiving signals from a receiver in an undergroundformation in response to signals generated from three or moretransmitting sources, each of the three or more transmitting sourceslocated at a known position, at least one transmitting source of thethree or more transmitting sources separated from and mounted on astructure different from at least one other transmitting source of thethree or more transmitting sources; determining angles with respect tothe transmitters relative to the receiver based on the received signals;and determining a position of the receiver based on the angles and theknown positions. The determining of angles and the determining of theposition of the receiver can be conducted downhole. Determining anglesand determining of the position of the receiver can be conducted usingelectronics integrated with the receiver, where the informationconcerning the known locations of the transmitting sensors is storedwith the electronics along with instructions to process the signals. Thedownhole processing can be located using electronics disposed on thestructure on which the receiver is disposed and separated from thereceiver. The downhole processing can allow for automated geosteering.Alternatively, the processing unit can be conducted at the surface inresponse to receiving the signals or data regarding the signals from thereceiver.

The second method can include controlling the three or more transmittingsources including a transmitting source that has a current-carrying wireof a closed loop of a circuit, the current-carrying wire being at aknown position and arranged along a straight-line path such that signalsreceived at the receiver from the closed loop are negligible fromportions of the closed loop that follow a path different from thestraight-line path. Features of the second method can includecontrolling the three or more transmitting sources including atransmitting source that has a number of current-carrying wires forminga closed loop of a circuit, each of the number of current-carrying wiresbeing at a known position and arranged along a straight-line path suchthat signals received at the receiver from the closed loop arenegligible from portions of the closed loop that follow a path differentfrom the straight-line paths; and processing the signals based on amodel of the number of current-carrying wires and their correspondingstraight-line paths. Controlling the three or more transmitting sourcescan include at least three dipole transmitters. The three or moretransmitting sources can include one or more transmitting sourceslocated aboveground. The second method can include generating at leastone signal of the generated signals from a transmitting sourceaboveground or near ground, the signal having a low frequency topenetrate deeply underground such that the signal is measurable in anunderground volume extending from a hundred feet to thousands feet indepth and from a hundred feet to thousands of feet across the depth.

The second method can include using gravity to provide a reference.Determining the position of the receiver can include evaluatinggeometric identities using the angles and the known positions.Evaluating geometric identities can include using a cosine theorem. Thesecond method can include operating each of the transmitters at afrequency different from that of the other ones of the number oftransmitters. The transmitters can also be operated sequentially.

In various embodiments, features of a third method to locate a receiverdownhole comprise: determining an electric field at a receiver, locatedin an underground formation, in response to signals generated from threeor more magnetic dipoles located at known positions such that there areat least three distinct planes defined respectively by location of oneof the three or more magnetic dipoles and the electric field at thereceiver due to the respective magnetic dipole; and determining theposition of the receiver based on the known positions and a direction ofthe electric field. Determining of the position of the receiver can beconducted downhole. Determining of the position of the receiver can beconducted using electronics integrated with the receiver, where theinformation concerning the known locations of the transmitting sensorsis stored with the electronics along with instructions to process thesignals. The downhole processing can be located using electronicsdisposed on the structure on which the receiver is disposed andseparated from the receiver. The downhole processing can allow forautomated geosteering. Alternatively, the processing unit can beconducted at the surface in response to receiving the signals or dataregarding the signals from the receiver.

In embodiments of the third method, the three or more magnetic dipolescan be located aboveground or near ground. In an embodiment, no morethan two transmitters and the receiver are in a plane.

In various embodiments, components of a system operable to find aposition in an underground formation, as described herein or in asimilar manner, can be realized in combinations of hardware and softwarebased implementations. These implementations can include amachine-readable storage device having machine-executable instructions,such as a computer-readable storage device having computer-executableinstructions, to find a position in an underground formation. Executedinstructions can also include instructions to operate one or moretransmitters to generate signals. Executed instructions can also includeinstructions to operate one or more receivers to provide signals inresponse to the signals generated by the one or more transmitters inaccordance with the teachings herein. The instructions can includeinstructions to provide data to a processing unit such that theprocessing unit conducts one or more processes to evaluate signals,data, or signals and data. Further, a machine-readable storage device,herein, is a physical device that stores data represented by physicalstructure within the device. Examples of machine-readable storagedevices include, but are not limited to, read only memory (ROM), randomaccess memory (RAM), a magnetic disk storage device, an optical storagedevice, a flash memory, and other electronic, magnetic, and/or opticalmemory devices.

In various embodiments, features of an embodiment of a machine-readablestorage device can include having instructions stored thereon, which,when performed by a machine, cause the machine to perform operations to:receive signals from a receiver in an underground formation in responseto signals generated from three or more transmitting sources, each ofthe three or more transmitting sources located at a known position, atleast one transmitting source of the three or more transmitting sourcesseparated from and mounted on a structure different from at least oneother transmitting source of the three or more transmitting sources; andprocess the received signals, using an inversion process based on thesignals generated from the three or more transmitting sources, todetermine the position of the receiver. The instructions can includeinstructions to control the three or more transmitting sources includinga transmitting source that has a current-carrying wire of a closed loopof a circuit, the current-carrying wire being at a known position andarranged along a straight-line path such that signals received at thereceiver from the closed loop are negligible from portions of the closedloop that follow a path different from the straight-line path. Theinstructions can include instructions to: control the three or moretransmitting sources including a transmitting source that has a numberof current-carrying wires forming a closed loop of a circuit, each ofthe number of current-carrying wires arranged along a straight-line pathsuch that signals received at the receiver from the closed loop arenegligible from portions of the closed loop that follow a path differentfrom the straight-line paths; and process the signals based on a modelof the number of current-carrying wires and their correspondingstraight-line paths. The instructions can include instructions tocontrol the three or more transmitting sources including at least threedipole transmitters. The instructions can include instructions tocontrol the three or more transmitting sources including a sourcegenerating large distribution of current aboveground or near ground thatgenerate electromagnetic fields below ground, the electromagnetic fieldsmeasurable at the receiver, the large distribution of current being at aknown position. The three or more transmitting sources can include nomore than two transmitters in a plane that contains the receiver. Thethree or more transmitting sources can include a transmitter in a well.The well can be different from a well in which the receiver is locatedor the well can be the well in which the receiver is located.Instructions controlling the three or more transmitting sources caninclude conducting various combinations of these features to control thethree or more transmitting sources.

The instructions can include instructions to generate at least onesignal of the generated signals from a transmitting source abovegroundor near ground, the signal having a low frequency to penetrate deeplyunderground such that the signal is measurable in an underground volumeextending from a hundred feet to thousands feet in depth and from ahundred feet to thousands of feet across the depth. The instructions caninclude instructions to operate the three or more transmitting sourcessequentially such that only one of the three or more transmittingsources is on at one time period. The instructions can includeinstructions to operate one or more of the transmitting sources locatedaboveground. The machine-readable storage device can includeinstructions to operate each of the transmitters at a frequency lessthan about 50 Hz.

In the instructions stored in the machine-readable storage device, usingthe inversion process can include: generating values of a signalexpected at the receiver from each of the transmitting sources;generating a difference between the signal expected and the signalreceived from the receiver; when the difference is less than athreshold, selecting values of coordinates for the receiver, as theposition of the receiver, that generated the signal expected at thereceiver for which the difference is less than the threshold; and whenthe difference is greater than the threshold, generating new values of asignal expected at the receiver and determine if a difference betweenthe new values and the signal received from the receiver is less thanthe threshold. Generating values of the signal expected at the receivercan include using an estimate of the position of the receiver with aforward model. Generating values of the signal expected at the receivercan include using an estimate of the position of the receiver with alookup table.

In the instructions stored in the machine-readable storage device, usingthe inversion process can include: generating an estimate of theposition of the receiver, the receiver taken as a first receiver;generating an estimate of each position of one or more other receivers,the one or more other receivers having a known position with respect tothe first receiver; generating values of signals expected at the firstreceiver and at the one or more other receivers from each of thetransmitting sources; generating a difference between the values of thesignals expected and a combination of the signal received at the firstreceiver and signals received at the one or more other receivers; whenthe difference is less than a threshold, selecting values of coordinatesof the first receiver, as the position of the first receiver, thatgenerated the signal expected at the first receiver for which thedifference is less than the threshold; and when the difference isgreater than the threshold, generating a new estimate of the position ofthe first receiver, if the inversion process is within a maximumiteration.

In the instructions stored in the machine-readable storage device, usingthe inversion process can include: generating sets of values of a signalexpected at the receiver, each set generated from a different estimateof the position of the receiver; generating differences between thevalues of the signal expected and the signal received from the receiverfor each set; selecting the estimate that minimizes error in thedifference between the values of the signal expected and the signalreceived from the receiver. Generating the sets of values of the signalexpected at the receiver can include using a forward model with each ofthe estimates.

In various embodiments, features of a second embodiment of amachine-readable storage device can include having instructions storedthereon, which, when performed by a machine, cause the machine toperform operations to: receive signals from a receiver in an undergroundformation in response to signals generated from three or moretransmitting sources, each of the three or more transmitting sourceslocated at a known position, at least one transmitting source of thethree or more transmitting sources separated from and mounted on astructure different from at least one other transmitting source of thethree or more transmitting sources; determine angles with respect to thetransmitters relative to the receiver based on the received signals; anddetermine the position of the receiver based on the angles and the knownpositions. The instructions can include instructions to control thethree or more transmitting sources including a transmitting source thathas a current-carrying wire of a closed loop of a circuit, thecurrent-carrying wire being at a known position and arranged along astraight-line path such that signals received at the receiver from theclosed loop are negligible from portions of the closed loop that followa path different from the straight-line path. The instructions caninclude instructions to: control the three or more transmitting sourcesincluding a transmitting source that has a number of current-carryingwires forming a closed loop of a circuit, each of the number ofcurrent-carrying wires being at a known position and arranged along astraight-line path such that signals received at the receiver from theclosed loop are negligible from portions of the closed loop that followa path different from the straight-line paths; and process the signalsbased on a model of the number of current-carrying wires and theircorresponding straight-line paths.

The instructions can include instructions to control the three or moretransmitting sources including at least three dipole transmitters. Theinstructions can include instructions to control the three or moretransmitting sources including transmitting sources located aboveground.The instructions can include instructions to generate at least onesignal of the generated signals from a transmitting source abovegroundor near ground, the signal having a low frequency to penetrate deeplyunderground such that the signal is measurable in an underground volumeextending from a hundred feet to thousands feet in depth and from ahundred feet to thousands of feet across the depth.

In the second embodiment of a machine-readable storage device caninclude instructions to perform operations, wherein operations todetermine the position can include evaluating geometric identities usingthe angles and the known positions. Evaluating geometric identities caninclude using a cosine theorem. The operations can include using gravityto provide a reference direction. The operations can include operatingeach of the transmitters at a frequency different from that of the otherones of the number of transmitters. The operations can include operatingthe transmitters sequentially with a single frequency.

In various embodiments, features of an embodiment of a machine-readablestorage device can include having instructions stored thereon, which,when performed by a machine, cause the machine to perform operations to:determine an electric field at a receiver, located in an undergroundformation, in response to signals generated from three or more magneticdipoles located at known positions such that there are at least threedistinct planes defined respectively by location of one of the three ormore magnetic dipoles and the electric field at the receiver due to therespective magnetic dipole; and determine the position of the receiverbased on the known positions and a direction of the electric field. Thethree or more magnetic dipoles can be located aboveground or nearground. The three or more magnetic dipoles can be arranged with no morethan two transmitters and the receiver in a plane.

In various embodiments, an embodiment of an example system can comprise:three or more transmitting sources, each of the sources located at aknown position, at least one transmitting source of the three or moretransmitting sources separated from and mounted on a structure differentfrom at least one other transmitting source of the three or moretransmitting sources; a control unit arranged to control generation ofsignals from the three or more transmitting sources; a receiver in anunderground formation, the receiver operable to receive signals inresponse to the generation from the three or more transmitting sources;and a processing unit arranged to process the received signals, using aninversion process based on the signals generated from the three or moretransmitting sources, to determine the position of the receiver. Theprocessing unit can be located downhole. The processing unit can berealized by electronics integrated with the receiver, where theinformation concerning the known locations of the transmitting sensorsis stored with the electronics along with instructions to process thesignals. The processing unit can be realized by electronics disposed onthe structure on which the receiver is disposed and separated from thereceiver. The processing unit can be located downhole, which can allowfor automated geosteering. Alternatively, the processing unit can belocated at the surface, responsive to receiving the signals or dataregarding the signals from the receiver.

The three or more transmitting sources can include a transmitting sourcethat has a current-carrying wire of a closed loop of a circuit, thecurrent-carrying wire being at a known position and arranged along astraight-line path such that signals received at the receiver from theclosed loop are negligible from portions of the closed loop that followa path different from the straight-line path. The three or moretransmitting sources can include a transmitting source that has a numberof current-carrying wires forming a closed loop of a circuit, each ofthe number of current-carrying wires being at a known position andarranged along a straight-line path such that signals received at thereceiver from the closed loop are negligible from portions of the closedloop that follow a path different from the straight-line paths; and theprocessing unit is arranged to process the signals based on a model ofthe number of current-carrying wires and their correspondingstraight-line paths. The three or more transmitting sources can includea source operable to generate large distribution of current abovegroundor near ground that generates electromagnetic fields below ground, theelectromagnetic fields measurable at the receiver, the largedistribution of current being at a known position. The three or moretransmitting sources can include one or more transmitting sourceslocated aboveground. The three or more transmitting sources can includea transmitter in a well. The well can be different from a well in whichthe receiver is located or the well can be the well in which thereceiver is located. The three or more transmitting sources can includeno more than two transmitters in a plane that contains the receiver. Thethree or more transmitting sources can be arranged with variouscombinations of these example embodiments of three or more transmittingsources.

The control unit can be structured to be operable to generate at leastone signal of the generated signals from a transmitting sourceaboveground or near ground, the signal having a low frequency topenetrate deeply underground such that the signal is measurable in anunderground volume extending from a hundred feet to thousands feet indepth and from a hundred feet to thousands of feet across the depth. Thecontrol unit can be structured to operate the three or more transmittingsources sequentially such that only one of the three or moretransmitting sources is on at one time period. The control unit can bearranged to operate each of the transmitting sources at a frequency lessthan about 50 Hz.

The processing unit can be arranged to use an inversion process thatincludes the processing unit operable to: generate values of a signalexpected at the receiver from each of the transmitting sources; generatea difference between the signal expected and the signal received fromthe receiver; when the difference is less than a threshold, selectvalues of coordinates for the receiver, as the position of the receiver,that generated the signal expected at the receiver for which thedifference is less than the threshold; and when the difference isgreater than the threshold, generate new values of a signal expected atthe receiver and determine if a difference between the new values andthe signal received from the receiver is less than the threshold. Theprocessing unit can be structured to be operable to use an estimate ofthe position of the receiver with a forward model. The processing unitcan be structured to be operable to use an estimate of the position ofthe receiver with a lookup table.

The processing unit can be arranged to use an inversion process thatincludes the processing unit operable to: generate an estimate of theposition of the receiver, the receiver taken as a first receiver;generate an estimate of each position of one or more other receivers,each of the one or more other receivers having a known position withrespect to the first receiver; generate values of signals expected atthe first receiver and at the one or more other receivers from each ofthe transmitting sources; generate a difference between the values ofthe signals expected and a combination of the signal received at thefirst receiver and signals received at the one or more other receivers;when the difference is less than a threshold, select values ofcoordinates of the first receiver, as the position of the firstreceiver, that generated the signal expected at the first receiver forwhich the difference is less than the threshold; and when the differenceis greater than the threshold, generate a new estimate of the positionof the first receiver, if the inversion process is within a maximumiteration.

The processing unit can be arranged to use an inversion process thatincludes the processing unit operable to: generate sets of values of asignal expected at the receiver, each set generated from a differentestimate of the position of the receiver; generate differences betweenthe values of the signal expected and the signal received from thereceiver for each set; and select the estimate that minimizes error inthe difference between the values of the signal expected and the signalreceived from the receiver. The processing unit can be structured to beoperable to use a forward model with each of the estimates.

In various embodiments, a second example of a system can comprise: threeor more transmitting sources, each of the transmitting sources locatedat a known position, at least one transmitting source of the three ormore transmitting sources separated from and mounted on a structuredifferent from at least one other transmitting source of the three ormore transmitting sources; a control unit arranged to control generationof signals from the three or more transmitting sources; a receiver in anunderground formation, the receiver operable to receive signals inresponse to the generation from the three or more transmitting sources;a processing unit arranged to determine angles with respect to thetransmitters relative to the receiver based on the received signals andto determine the position of the receiver based on the angles and theknown positions. The processing unit can be located downhole. Theprocessing unit can be realized by electronics integrated with thereceiver, where the information concerning the known locations of thetransmitting sensors is stored with the electronics along withinstructions to process the signals. The processing unit can be realizedby electronics disposed on the structure on which the receiver isdisposed and separated from the receiver. The processing unit can belocated downhole, which can allow for automated geosteering.Alternatively, the processing unit can be located at the surface,responsive to receiving the signals or data regarding the signals fromthe receiver.

In the second embodiment of an example system, the three or moretransmitting sources can include a transmitting source that has acurrent-carrying wire of a closed loop of a circuit, thecurrent-carrying wire being at a known position and arranged along astraight-line path such that signals received at the receiver from theclosed loop are negligible from portions of the closed loop that followa path different from the straight-line path. The three or moretransmitting sources can include a transmitting source that has a numberof current-carrying wires forming a closed loop of a circuit, each ofthe number of current-carrying wires being at a known position andarranged along a straight-line path such that signals received at thereceiver from the closed loop are negligible from portions of the closedloop that follow a path different from the straight-line paths; and theprocessing unit is arranged to process the signals based on a model ofthe number of current-carrying wires and their correspondingstraight-line paths. The three or more transmitting sources can includeat least three dipole transmitters. The three or more transmittingsources can include one or more transmitting sources locatedaboveground.

The control unit can be structured to be operable to generate at leastone signal of the generated signals from a transmitting sourceaboveground or near ground, the signal having a low frequency topenetrate deeply underground such that the signal is measurable in anunderground volume extending from a hundred feet to thousands feet indepth and from a hundred feet to thousands of feet across the depth. Thecontrol unit can be arranged to operate each of the transmitters at afrequency different from that of the other ones of the number oftransmitters. The transmitters can also be operated sequentially.

In the second embodiment of an example system, the processing unit canbe arranged to evaluate geometric identities using the angles and theknown positions. The processing unit can be arranged to use a cosinetheorem to evaluate geometric identities. The processing unit can bestructured to operate to use gravity to provide a reference direction.

In various embodiments, an embodiment of a third example system cancomprise: a receiver located in an underground formation; three or moremagnetic dipoles located at known positions such that there are at leastthree distinct planes defined respectively by location of one of thethree or more magnetic dipoles and the electric field at the receiverdue to the respective magnetic dipole; a control unit arranged tocontrol generation of signals from the three or more magnetic dipoles;and a processing unit arranged to determine an electric field at thereceiver in response to generating the signals, and to determine theposition of the receiver based on the known positions and the directionof the electric field. The processing unit can be located downhole. Theprocessing unit can be realized by electronics integrated with thereceiver, where the information concerning the known locations of thetransmitting sensors is stored with the electronics along withinstructions to process the signals. The processing unit can be realizedby electronics disposed on the structure on which the receiver isdisposed and separated from the receiver. The processing unit can belocated downhole, which can allow for automated geosteering.Alternatively, the processing unit can be located at the surface,responsive to receiving the signals or data regarding the signals fromthe receiver.

In the third embodiment of an example system, the three or more magneticdipoles are located aboveground or near ground. In an embodiment, anarrangement can include no more than two magnetic dipoles and thereceiver in a plane.

Permutations of features of the methods discussed herein can be realizedamong the different methods. Permutations of features of themachine-readable storage devices discussed herein can be realized amongthe different machine-readable storage devices. Permutations of featuresof the systems discussed herein can be realized among the differentsystems.

FIG. 23 depicts a block diagram of features of an example embodiment ofa system 2300 operable to find a position in an underground formation.System 2300 configured with one or more transmitting sensors located atknown positions and one or more receivers located in the undergroundformation. System 2300 includes an arrangement of transmitting sensors2312 and receiving sensors 2310 that can be realized in a similar oridentical manner to arrangements of sensors discussed herein. System2300 can be configured to operate in accordance with the teachingsherein.

System 2300 can include a controller 2325, a memory 2330, an electronicapparatus 2365, and a communications unit 2335. Controller 2325, memory2330, and communications unit 2335 can be arranged to operate as acontrol unit and a processing unit to control operation of thearrangement of transmitting sensors 2312 and receiving sensors 2310 andto perform one or more processing operations on the signals collected todetermine the position of one or more of the receiving sensors 2310, ina manner similar or identical to the procedures discussed herein. Such aprocessing unit can be realized as a processing unit 2320 that can beimplemented as a single unit or distributed among the components ofsystem 2300 including electronic apparatus 2365. Controller 2325 andmemory 2330 can operate to control activation of transmitter sensors2312 and selection of receiver sensors 2310 and to manage processingschemes in accordance with measurement procedures and signal processingas described herein. System 2300 can be structured to function in amanner similar to or identical to structures associated withtransmitting arrangements and methods of processing a signal or signalsfrom a receiving unit, whose position can be determined by theprocessing.

Communications unit 2335 can include downhole communications forappropriately located sensors. Such downhole communications can includea telemetry system. Communications unit 2335 may use combinations ofwired communication technologies and wireless technologies atfrequencies that do not interfere with on-going measurements.Communications unit 2335 can include interfaces to communicate withtransmitting sensors distributed over a large spatial region.

System 2300 can include a network 2327, where network 2327 is operableover a network providing electrical conductivity among subsystems ofsystem 2300 distributed over a large spatial region including surfacelocated transmitters, underground transmitters, and receivers inunderground formations. The surface located transmitters and theunderground transmitters can be located at known locations with one ormore receivers in communication to provide signals to processing unit tofind the position of one or more receivers. Network 2327 can include anaddress bus, a data bus, and a control bus, each independentlyconfigured or in an integrated format. Network 2327 can be realizedusing a number of different communication mediums that allows forcontrol and management of components and subsystems of system 2300 thatcan be distributed over a large spatial region. Use of network 2327 canbe regulated by controller 2325.

In various embodiments, peripheral devices 2345 can include additionalstorage memory and/or other control devices that may operate inconjunction with controller 2325 and/or memory 2330. In an embodiment,controller 2325 is realized as a processor or a group of processors thatmay operate independently depending on an assigned function. Peripheraldevices 2345 can be arranged with one or more displays 2355, as adistributed component on the surface, which can be used withinstructions stored in memory 2330 to implement a user interface tomonitor the operation of system 2300 and/or components distributedwithin system 2300. The user interface can be used to input operatingparameter values such that system 2300 can operate autonomouslysubstantially without user intervention.

Although specific embodiments have been illustrated and describedherein, it will be appreciated by those of ordinary skill in the artthat any arrangement that is calculated to achieve the same purpose maybe substituted for the specific embodiments shown. Various embodimentsuse permutations and/or combinations of embodiments described herein. Itis to be understood that the above description is intended to beillustrative, and not restrictive, and that the phraseology orterminology employed herein is for the purpose of description.Combinations of the above embodiments and other embodiments will beapparent to those of skill in the art upon studying the abovedescription.

What is claimed is:
 1. A method comprising: receiving signals from areceiver in an underground formation in response to signals generatedfrom three or more transmitting sources, each of the three or moretransmitting sources located at a known position, at least onetransmitting source of the three or more transmitting sources separatedfrom and mounted on a structure different from at least one othertransmitting source of the three or more transmitting sources; using aninversion process to (i) determine angles with respect to thetransmitters relative to the receiver based on the received signals; and(ii) determine a position of the receiver based on the angles and theknown positions wherein using the inversion process further comprisesgenerating values of signals expected at the receiver from each of thethree or more transmitting sources; comparing the values of the signalsexpected and the signals received by the receiver to (i) determine theangles with respect to the transmitters relative to the receiver basedon the comparison; and (ii) determine the position of the receiver basedon the comparison.
 2. The method of claim 1, wherein the method includescontrolling the three or more transmitting sources including atransmitting source that has a current-carrying wire of a closed loop ofa circuit, the current-carrying wire being at a known position, thecurrent-carrying wire arranged along a straight-line path such thatsignals received at the receiver from the closed loop are negligiblefrom portions of the closed loop that follow a path different from thestraight-line path.
 3. The method of claim 1, wherein the methodincludes controlling the three or more transmitting sources including atransmitting source that has a number of current-carrying wires forminga closed loop of a circuit, each of the number of current-carrying wiresbeing at a known position and arranged along a straight-line path suchthat signals received at the receiver from the closed loop arenegligible from portions of the closed loop that follow a path differentfrom the straight-line paths; and processing the signals based on amodel of the number of current-carrying wires and their correspondingstraight-line paths.
 4. The method of claim 1, wherein the methodincludes controlling the three or more transmitting sources including atleast three dipole transmitters.
 5. The method of claim 1, wherein thethree or more transmitting sources include one or more transmittingsources located aboveground.
 6. The method of claim 1, wherein themethod includes generating at least one signal of the generated signalsfrom a transmitting source aboveground or near ground, the signal havinga low frequency to penetrate deeply underground such that the signal ismeasurable in an underground volume extending from a hundred feet tothousands of feet in depth and from a hundred feet to thousands of feetacross the depth.
 7. The method of claim 1, wherein determining theposition includes evaluating geometric identities using the anglesdetermined via the inversion process and the known positions.
 8. Themethod of claim 7, wherein evaluating geometric identities includesusing a cosine theorem.
 9. The method of claim 1, wherein the methodincludes using gravity to provide a reference.
 10. The method of claim1, wherein the method includes operating each of the transmitters at afrequency different from that of the other ones of the number oftransmitters.
 11. The method of claim 1, wherein the angles comprise anazimuth and elevation angle of the receiver.
 12. The method of claim 1,wherein a drilling operation is performed based, at least in part, onthe position and angle.
 13. A machine-readable storage device havinginstructions stored thereon, which, when performed by a machine, causethe machine to perform operations to: receive signals from a receiver inan underground formation in response to signals generated from three ormore transmitting sources, each of the three or more transmittingsources located at a known position, at least one transmitting source ofthe three or more transmitting sources separated from and mounted on astructure different from at least one other transmitting source of thethree or more transmitting sources; using an inversion process to (i)determine angles with respect to the transmitters relative to thereceiver based on the received signals; and (ii) determine a position ofthe receiver based on the angles and the known positions, wherein usingthe inversion process further comprises generating values of signalsexpected at the receiver from each of the three or more transmittingsources; and comparing the values of the signals expected and thesignals received by the receiver to (i) determine the angles withrespect to the transmitters relative to the receiver based on thecomparison; and (ii) determine the position of the receiver based on thecomparison.
 14. The machine-readable storage device of claim 13, whereinthe instructions include instructions to control the three or moretransmitting sources including a transmitting source that has acurrent-carrying wire of a closed loop of a circuit, thecurrent-carrying wire being at a known position, the current-carryingwire arranged along a straight-line path such that signals received atthe receiver from the closed loop are negligible from portions of theclosed loop that follow a path different from the straight-line path.15. The machine-readable storage device of claim 13, wherein theinstructions include instructions to: control the three or moretransmitting sources including a transmitting source that has a numberof current-carrying wires forming a closed loop of a circuit, each ofthe number of current-carrying wires being at a known position andarranged along a straight-line path such that signals received at thereceiver from the closed loop are negligible from portions of the closedloop that follow a path different from the straight-line paths; andprocess the signals based on a model of the number of current-carryingwires and their corresponding straight-line paths.
 16. Themachine-readable storage device of claim 13, wherein the instructionsinclude instructions to control the three or more transmitting sourcesincluding at least three dipole transmitters.
 17. The machine-readablestorage device of claim 13, wherein the instructions includeinstructions to control the three or more transmitting sources includingtransmitting sources located aboveground.
 18. The machine-readablestorage device of claim 13, wherein the instructions includeinstructions to generate at least one signal of the generated signalsfrom a transmitting source aboveground or near ground, the signal havinga low frequency to penetrate deeply underground such that the signal ismeasurable in an underground volume extending from a hundred feet tothousands of feet in depth and from a hundred feet to thousands of feetacross the depth.
 19. The machine-readable storage device of claim 13,wherein operations to determine the position include evaluatinggeometric identities using the angles determined via the inversionprocess and the known positions.
 20. The machine-readable storage deviceof claim 19, wherein evaluating geometric identities includes using acosine theorem.
 21. The machine-readable storage device of claim 13,wherein the operations include using gravity to provide a referencedirection.
 22. The machine-readable storage device of claim 13, whereinthe operations include operating each of the transmitters at a frequencydifferent from that of the other ones of the number of transmitters. 23.The machine-readable storage device of claim 13, wherein the operationsinclude operating the transmitters sequentially with a single frequency.24. The machine-readable storage device of claim 13, wherein the anglescomprise an azimuth and elevation angle of the receiver.
 25. Themachine-readable storage device of claim 13, wherein a drillingoperation is performed based, at least in part, on the position andangle.
 26. A system comprising: three or more transmitting sources, eachof the transmitting sources located at a known position, at least onetransmitting source of the three or more transmitting sources separatedfrom and mounted on a structure different from at least one othertransmitting source of the three or more transmitting sources; a controlunit arranged to control generation of signals from the three or moretransmitting sources; a receiver in an underground formation, thereceiver operable to receive signals in response to the generation fromthe three or more transmitting sources; a processing unit arranged touse an inversion process to (i) determine angles with respect to thetransmitters relative to the receiver based on the received signals; andto (ii) determine a position of the receiver based on the angles and theknown positions, wherein using the inversion process further comprisesgenerating values of signals expected at the receiver from each of thethree or more transmitting sources; comparing the values of the signalsexpected and the signals received by the receiver to (i) determine theangles with respect to the transmitters relative to the receiver basedon the comparison; and (ii) determine the position of the receiver basedon the comparison.
 27. The system of claim 26, wherein the three or moretransmitting sources include a transmitting source that has acurrent-carrying wire of a closed loop of a circuit, thecurrent-carrying wire being at a known position, the current-carryingwire arranged along a straight-line path such that signals received atthe receiver from the closed loop are negligible from portions of theclosed loop that follow a path different from the straight-line path.28. The system of claim 26, wherein the three or more transmittingsources include a transmitting source that has a number ofcurrent-carrying wires forming a closed loop of a circuit, each of thenumber of current-carrying wires being at a known position and arrangedalong a straight-line path such that signals received at the receiverfrom the closed loop are negligible from portions of the closed loopthat follow a path different from the straight-line paths; and theprocessing unit is arranged to process the signals based on a model ofthe number of current-carrying wires and their correspondingstraight-line paths.
 29. The system of claim 26, wherein the three ormore transmitting sources include at least three dipole transmitters.30. The system of claim 26, wherein the three or more transmittingsources include one or more transmitting sources located aboveground.31. The system of claim 26, wherein the control unit is operable togenerate at least one signal of the generated signals from atransmitting source aboveground or near ground, the signal having a lowfrequency to penetrate deeply underground such that the signal ismeasurable in an underground volume extending from a hundred feet tothousands of feet in depth and from a hundred feet to thousands of feetacross the depth.
 32. The system of claim 26, wherein the processingunit arranged to determine the position includes the processing unitarranged to evaluate geometric identities using the angles determinedvia the inversion process and the known positions.
 33. The system ofclaim 26, wherein the processing unit arranged to evaluate geometricidentities includes the processing unit arranged to use a cosinetheorem.
 34. The system of claim 26, wherein the processing unitincludes the processing unit operable to use gravity to provide areference direction.
 35. The system of claim 26, wherein the controlunit is arranged to operate each of the transmitters at a frequencydifferent from that of the other ones of the number of transmitters. 36.The system of claim 26, wherein the angles comprise an azimuth andelevation angle of the receiver.